Vershik, A. M.; Kerov, S. V. Characters and realizations of representations of an infinite-dimensional Hecke algebra, and knot invariants. (English. Russian original) Zbl 0716.20008 Sov. Math., Dokl. 38, No. 1, 134-137 (1989); translation from Dokl. Akad. Nauk SSSR 301, No. 4, 777-780 (1988). In this paper we obtain q-analogues of our results [in Funkts. Anal. Prilozh. 15, No.4, 15-27 (1981; Zbl 0507.20006) and Dokl. Akad. Nauk SSSR 257, 1037-1040 (1981; Zbl 0534.20008)] on characters and quotient representations of the symmetric group \({\mathfrak S}_{\infty}\) for an infinite-dimensional Hecke algebra \(H_{\infty}(q)\) and we realize its quotient representation, using certain solutions of the Yang-Baxter equation. As an application we deduce in a simple way some recent results of Jones and others on new knot invariants. Cited in 11 Documents MSC: 20C32 Representations of infinite symmetric groups 57M25 Knots and links in the \(3\)-sphere (MSC2010) 16S34 Group rings Keywords:characters; quotient representations; symmetric group; Hecke algebra; Yang-Baxter equation; knot invariants Citations:Zbl 0507.20006; Zbl 0534.20008 PDFBibTeX XMLCite \textit{A. M. Vershik} and \textit{S. V. Kerov}, Sov. Math., Dokl. 38, No. 1, 134--137 (1989; Zbl 0716.20008); translation from Dokl. Akad. Nauk SSSR 301, No. 4, 777--780 (1988)